
measures the average of two-point correlations of \(u\) : More precisely, if \(\mu ^\Delta _t\)  is of the form \(\mu ^\Delta _t = S^{\Delta }_{t,\#}\mu _0\) , \(\mu _0 \in {\mathcal {P}}_2(L^2_x)\) , with \(S^\Delta _t: L^2_x\rightarrow L^2_x\)  satisfying assumption REF , and if we have \({S}_2^T(\mu ^\Delta _t;r) \le \phi (r)\) , for some modulus of continuity \(\phi (r)\)  (i.e., \(\phi (r) > 0, ~\forall r\)  and \(\lim \limits _{r \rightarrow 0} \phi (r) = 0\) ), uniformly in \(\Delta \) , then \(\mu ^{\Delta }_t\)  is compact in \(L^1_t({\mathcal {P}})\) . The quantity \(r \mapsto {S}_2^T(\mu ^\Delta _t;r)\)  is referred to as the (time-integrated) structure function of \(\mu ^\Delta _t\) . For simplicity, we will state the following results in the periodic setting with domain \(D = d\) . Numerical evidence for the uniform boundedness of these structure functions for the statistical forward problem has been presented for a variety of initial probability measures \({\mu }\)  supported on rough initial data of the two-dimensional incompressible Euler equations in [1]}, [2]}, and in the context of hyperbolic conservation laws in [3]}.
